I agree with your definition - but only from a pro-am's perspective.
If you are an amateur couple then:
Instructor - trainer for beginners or chain-school
Teacher - to some an expert that trains you through syllabus; but others use it as a general term (often a long-term trainer)
Pro - professional couple (the term is irrelevant to teaching AMs since all teachers are pros)
Coach - could be your regular (regular) teacher or an expert you go to once or once in a while